Calculating Updated Sample Average

The problem asks us to find the new sample average after adding a new data point.

Initial Data Summary

• Initial number of data points ($n_1$): 50
• Initial sample average ($\bar{x}_1$): 40

Calculating Initial Sum

The sum of the initial data points can be calculated using the formula:

$ \text{Sum}_1 = n_1 \times \bar{x}_1 $

Substituting the values:

$ \text{Sum}_1 = 50 \times 40 = 2000 $

Updating Data and Sum

A new data point is added:

• New data point value: 142
• New number of data points ($n_2$): $50 + 1 = 51$

The new sum ($\text{Sum}_2$) is the initial sum plus the new data point:

$ \text{Sum}_2 = \text{Sum}_1 + 142 $

$ \text{Sum}_2 = 2000 + 142 = 2142 $

Calculating New Average

The updated sample average ($\bar{x}_2$) is calculated by dividing the new sum by the new number of data points:

$ \bar{x}_2 = \frac{\text{Sum}_2}{n_2} $

$ \bar{x}_2 = \frac{2142}{51} $

Performing the division:

$ \bar{x}_2 = 42 $

Conclusion

The updated sample average is exactly 42.